8. Angle Relationships, Similar Figures, and Special Triangles

Angle relationships:

• Supplementary: two angles summing to \(180^\circ\).
• Complementary: two angles summing to \(90^\circ\).
• Vertical angles: opposite angles at an intersection — always equal.
• Transversal angles: when a line crosses two parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles are supplementary.

Triangle angle sum: the three interior angles of any triangle sum to \(180^\circ\).

Exterior angle theorem: an exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Similar figures: same shape, different size. Corresponding angles are equal; corresponding sides are proportional.
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}. \]
Perimeter scales by the ratio; area scales by the square of the ratio.

Special right triangles (memorize these):

• 45-45-90: sides are \(x : x : x\sqrt{2}\). Legs equal; hypotenuse is leg times \(\sqrt{2}\).
• 30-60-90: sides are \(x : x\sqrt{3} : 2x\). The shorter leg is opposite \(30^\circ\); the hypotenuse is twice the shorter leg.